a Holes in a quantum spin liquid * + (a) + Guangyong Xu , G. Aeppli , M.E. Bisher , C. Broholm*,§, J. F. DiTusa, C. D. Frost#, T.Ito**, K. Oka**, H. Takagi++& M.M.J. Treacy+ * Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA + NEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540, USA § NIST Center for Neutron Research, Gaithersburg, Maryland 20899, USA || Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA # ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK ** Electrotechnical Laboratory, Tsukuba 305, Japan ++ The Institute of Solid State Physics, University of Tokyo, Roppongi, Tokyo 106 Japan Because of possible connections to high temperature superconductivity, spin density modulations in transition metal oxides are receiving huge attention. The modulations appear upon introduction of charge carriers, via chemical substitution, into an insulating and antiferromagnetic parent compound. They tend to be static when the carriers are frozen, dynamic when they are mobile. Until now, the evidence for such modulations has been largely confined to lamellar compounds1,2,3,4, i.e. materials whose magnetism and charge transport are essentially two-dimensional. It has also been restricted to materials whose parent insulators are ordered antiferromagnets. In this Letter, we provide the first evidence for analogous phenomena in a doped transition metal oxide chain compound, Y2-xCaxBaNiO5, for which the parent is a paramagnet by virtue of quantum fluctuations4,5,6,7. The key features of this orthorhombic material are the chains of NiO6 octahedra shown in Fig. 1(a). The octahedra are cornerrather than edge-sharing, which results in the dominance of the magnetism by the very simple ..O-Ni-O-Ni-O.. back-bone. The electrical conduction is also primarily along the chains, as indicated by the more than three orders of magnitude smaller resistivity recorded along the crystallographic a axis which is parallel to the chains, than along the perpendicular b and c axes8. The orbital moments on the Ni2+ ions are quenched, so that the magnetic degree of freedom at each Ni site is the spin S=1 associated with the d8 configuration of Ni2+. Each S=1 ion is coupled to its neighbors via antiferromagnetic super-exchange through intermediate O2- ions. Replacing the off-chain Y3+ ions by Ca2+ ions introduces holes primarily onto apical oxygen atoms in the chains, and increases the electrical conductivity at finite temperatures8. Thus, Y2-xCaxBaNiO5 is a one-dimensional analog of the cuprates, where off-(CuO)-plane chemical impurities donate holes to the CuO planes. Magnetic one-dimensionality causes the x=0 parent compound of the Y2-xCaxBaNiO5 series to be a paramagnet prevented from ordering antiferromagnetically by quantum fluctuations. The material is not an ordinary paramagnet with heavily damped spin fluctuations, but rather it is the magnetic analog of super-fluid 4He in the sense that it has a macroscopically coherent quantum ground state even while lacking classically discernible order. Affleck, Kennedy Lieb and Tasaki provided a prescription for this state which we have illustrated in Fig. 1(b) 9,10. They considered each S=1 degree of freedom as the triplet ground state formed between two ferromagnetically coupled S=1/2 degrees of freedom, in accord with atomic physics which posits a large and positive Hund’s rule coupling between the two S=1/2 Ni O Ca Y (b) (c) Figure 1. (a) Chain unit of Y2BaNiO5 extending along the a-direction and featuring nickel atoms with octahedral oxygen coordination and Ca2+ impurities on Y3+ sites. (b) Schematic of spin-1/2 degrees of freedom at the edges of the AKLT state. (c) Ferromagnetically coupled chain end spin-1/2 degrees of freedom. holes in the d-shell of Ni2+. The S=1 antiferromagnetic chain is then an S=1/2 chain with twice the number of sites and bonds of alternating sign. If we could tune the ferromagnetic Hund coupling to a value well below the antiferromagnetic interaction between different S=1 sites, the ground state would be a chain of singlets formed between S=1/2 spins on neighboring sites. A gap equal to the antiferromagnetic coupling would separate the ground state bond singlets from the excited state triplets. A large body of work performed over the last 15 years shows that the AKLT ground and excited states evolve adiabatically into the Haldane ground state as the ferromagnetic, or Hund’s rule, coupling is taken to much larger values than J. The major change is that for energies below the intra-atomic exchange energy, S=1/2 degrees of freedom are never observed singly (i.e. they are confined much as the fractional charges of quarks are confined), and that the triplet band exhibits dispersion, with a minimum energy, called the Haldane gap11 =0.41 J17. Fig 2(a) shows that the triplet band can be observed experimentally for Y2BaNiO5. The figure, an image of the magnetic fluctuations as function of wave-number (along the chains) Q and energy , was obtained using the multi-detector time-of-flight neutron scattering spectrometer MARI at the pulsed spallation source ISIS in the UK. Magnetic fluctuations are found only when and Q lie on a well-defined dispersion relation, which is direct evidence for a coherent propagating triplet mode. What distinguishes this mode from a spin wave is that it propagates in a medium with no static magnetic order-parameter. Key features are (i) the 9 meV gap first predicted by Haldane11, which corresponds to the threshold energy for triplet creation. (ii) The magnetic excitation bandwidth of 62 meV that translates into an antiferromagnetic exchange constant J=21 meV, and (iii) the vanishing intensity for Q=n2 that is evidence for an isotropic singlet ground state. What happens when we add carriers by Ca doping the quantum spin liquid of Fig 2 (a)? Fig 2 (b), collected for our x 0.095 0.005 sample, reveals excitations following the same underlying dispersion relation as for pure Y2BaNiO5. However near the former minimum at Q= and =9 meV, their intensity is substantially reduced and new magnetic scattering appears below the Haldane gap. Fig. 3 (b) shows more detailed data (a) (b) Figure 2. Coarse-resolution overview of the magnetic fluctuations in (a) pure and (b) 9.5% doped Y2BaNiO5 measured at T=10K. The initial neutron beam energy was 90 meV and the sample was oriented with its chain axis perpendicular to the incident beam direction. The boxes indicate the region of the energy-wave-vector phase space examined in the high-resolution survey of Fig.3 collected near and below the gap. Near the gap energy, the scattering is clearly peaked at the commensurate Q= position but in the 3-7 meV range it is most intense along two vertical lines, displaced symmetrically to the left and right of Q=. In addition to the energy-resolved data of Fig.3, we have obtained higher statistics energy-integrated Q-scans to better quantify the incommensurate peaks. Fig. 4 shows the results for two doped single crystals. Increased doping results in a peak splitting which is only slightly larger for the x=0.14 sample. Specifically, if we fit both data sets by the sum of two Lorentzians centered at with equal amplitudes and half-widths-at-halfmaximum , we find =0.085(3) and 0.095(4), and =0.071(4) and 0.072(5) for x=0.095(5) and 0.14(1) respectively. Thus, the wavelength / associated with the incommensurability, is 10% more than the inter-impurity spacing (a) (b) Figure 3. Low energy detail of magnetic excitations in (a) pure and (b) 9.5%doped Y2BaNiO5. (a) shows MARI data at T=10K while (b) shows data measured at T=1.5K, and the spectrometers used were SPINS and BT2 at NIST with final neutron energies fixed at 5 and 14.7 meV respectively. Figure 4. Sweeps as a function of wave-vector through the incommensurate peaks at fixed energy transfer of 4.6meV for two different hole concentrations. The energy resolution of the spectrometer was 2meV FWHM. The yellow line in the top frame shows the single impurity model Eq. (1). The red lines take into account that neighboring impurities truncate the spin polarization cloud around an impurity bond. The green lines show the dual-Lorentzian model. 1/x for x=0.095 and 40% more for x=0.14. Nonetheless / is of the same order as several other length scales, namely the magnetic correlation length for the parent compound, the magnetic coherence length, =1/, for the incommensurate fluctuations themselves, and the average impurity spacing 1/x. There are various models, which will produce incommensurate fluctuations of the type, which we have discovered in Y2xCaxBaNiO5. The first proceeds from a charge-ordering hypothesis where the holes order periodically to minimize their mutual electrostatic repulsion and the spin system acquires a period commensurate to that of the hole lattice. The magnetic fluctuations are then spin waves broadened as a function of wave-number because the hole lattice is somewhat disordered. Such a scenario is commonly supposed for the two-dimensional analog of Y2xCaxBaNiO5, namely doped La2NiO4, and key experimental support is derived from electron12 and neutron diffraction13,14 measurements, which revealed the lattice distortions, which one might associate with hole ordering. Thus inspired, we have used electron diffraction to search for hole ordering in Y2-xCaxBaNiO5 down to 15K. The failure of this search and the absence of proportionality between x and predicted by such a model, make the charge ordering explanation of our experiments unlikely. A second possibility is to identify the incommensurate magnetic fluctuations with electron-hole pair excitations in the one-dimensional hole liquid associated with mobile S=1/2 holes dressed by their propagation through the Haldane paramagnet15. The wave-vectors would be vectors spanning the Fermi surface while the vertical nature of the incommensurate streaks in Fig. 3 would imply a Fermi velocity in excess of 0.5 eVÅngstrom. The problem with this picture is not so much that the sample resistitivities are high at low T – the resistivity data could well be consistent with metallic behavior on the scale of the Haldane length - as that the Fermi sea is expected to shrink in proportion to x, a result again contradicted by the almost negligible increase in on raising x from 0.095 to 0.14. A last possibility starts from consideration16 of the magnetic structure factor for a single static hole donated to the NiO chain by the Ca ions. Because it carries a spin, a localized hole on a super- exchange mediating oxygen atom favors parallel alignment of neighboring Ni spins and hence induces an effective ferromagnetic nearest neighbor interaction. Thus, the infinite chain is divided into two semi-infinite segments, and the ground and excited state wavefunctions can be constructed from a basis set which is the direct product of wavefunctions for the left and right segments and for the hole on the intervening oxygen. The ground state wavefunction for a semi-infinite segment is a doublet rather than the singlet wavefunction for the unterminated infinite chain. Fig. 1(b)-(c) give a pictorial justification: the chain ends have unpaired spins because only one of the two AKLT spins which together represent the S=1 sites can be paired with an AKLT spin on a neighboring spin S=1 site. As we move from the alternating valence bond representation to the real S=1 chain, the doublet wavefunction is not confined to the chain end, but extends into the chain over a distance of order the finite magnetic correlation length associated with the Haldane gap. A ferromagnetic bond inserted between two semi-infinite chains combines the doublets found on either side of the bond into a triplet, again with spatial extent of order the Haldane length. Therefore, we can describe the spin polarization around the impurity as antiferromagnetic with a decay j envelope, i.e. S0 S j 1 j e . Neutron scattering measures the Fourier transform of the spin polarization, which for the triplet ground state associated with a single ferromagnetic bond, is S (Q ) F (Q ) (1 e ) cos Q / 2 cosh cos Q 2 (1) Eq. (1) describes a function with nodes at odd multiples of and pairs of peaks, with half widths and incommensurability of order . Thus, the description leading to Eq. (1) does have a chance of accounting for a key feature of our experiments. Even so, there are two clear difficulties, the first being that the nodes predicted by (1) are not seen experimentally. The second problem is that because Eq. (1) describes the ground state for a single ferromagnetic bond, neutron scattering should not involve any energy loss, implying that the scattering should be elastic rather than inelastic as seen experimentally. The first difficulty has a simple resolution. At finite hole (and therefore ferromagnetic bond) densities, the polarization clouds associated with the holes overlap and the expression (1), derived for isolated impurities, becomes invalid. A crude model, which considers the overlaps, simply truncates the polarization clouds at the neighboring impurity sites. Because the bonds are randomly distributed, the inversion symmetry characterizing isolated impurities is broken and intensity becomes allowed at Q=. The only parameters in such a description are the FM bond density and, as for Eq (1), the extent of the polarization cloud 1/. The first is equated with the measured Ca concentration. We adjusted the second to optimize the fit of the corresponding structure factor to our data. We also allowed for a spin on the oxygen atom whose main effect is to change the relative intensity of the two peaks. Best agreement with the data was obtained for O / Ni 0.1(1) where Ni is the defect induced moment on the chain end Ni site. As shown by the red lines in Fig. 4 the model provides a good account of the data for = -1 =7.5(4) and 7.4(4) for the two concentrations. Both values for are close to the equal time spin correlation length of the spin-1 chain which is 6.03 lattice spacings17. How do we account for the inelasticity of the incommensurate signal? One approach is to view the chain as consisting not of the original S=1 degrees of freedom from the Ni2+ ions but of the effective spin degrees of freedom associated with the randomly placed holes. The latter interact via the overlapping tails of their antiferromagnetic polarization clouds, which will imply effective couplings of random sign (because the distances between two holes are as probable to be even as odd multiples of the Ni-Ni separation) of typical magnitude ~4meV. Neglecting quantum fluctuations, the ground state is likely to be a spin glass, as has deduced from other experiments18 on Y2-xCaxBaNiO5. Furthermore, we expect to see spin excitations above such a ground state with a bandwidth ~zS<|J’|> which evaluates to 8 meV. The incommensurate nature of the excitations simply follows from the structure factor of the composite spin induced by a hole and should dissolve only when these effective degrees of freedom are themselves dissolved as either the Haldane gap energy or the ferromagnetic bond strength is exceeded. By integrating the data for our x=0.095 sample over Q and we derive the total squared moment associated with a ferromagnetic impurity bond in the spin-1 chain to be 7 2 B2. This number is close to the total spectral weight associated with the ground state triplet of a ferromagnetically coupled spin-1/2 pair, which is 8 B2. Note however that because we are dealing with composite chain end degrees of freedom there is in fact no simple reason that these numbers should be the same even if the impurity model we are discussing is correct. In summary, we have performed the first measurements of the magnetic fluctuations in single crystals of a doped spin-1 chain with antiferromagnetic couplings. At high energies, well above the Haldane gap the triplet excitations associated with the spin liquid ground state of the parent compound persist with doping. However, below the Haldane gap, new excitations appear with a broad spectrum and characteristic wave vectors, which are, displaced from the zone boundary by an amount of order the inverse correlation-length in the bulk spin chains. Our results are significant beyond the nickel chain compound we have studied. This follows from the strong similarities of our data to those for the superconducting cuprates, where we also see incommensurate fluctuations at low energies and remnant spin-wave-like signals at high energies. The one-dimensionality makes Y2BaNiO5 much easier to model and our analysis reveals that “incommensurate” peaks arise naturally even without hole order because of the characteristic spin density modulation that develops around a defect in the singlet ground state of a quantum magnet. This phenomenon accounts for the saturation of the incommensurability with increased doping in the one-dimensional nickelate, and could account for the same phenomenon in the two-dimensional cuprates. Indeed, the one-dimensional nickelate gives us our first quantitative impression of the magnetic polarization cloud associated with the holes in a doped transition metal oxide. Our results imply that the spin part of the hole wave-function is actually the edge state nucleated by the hole in a quantum spin fluid. 1 Cheong, S.-W. et al. Incommensurate magnetic fluctuations in La2-xSrxCuO4. Phys. Rev. Lett. 67,1791-1794 (1991). 2 Mook, H. A. et al. Spin fluctuations in YBa2Cu3O6.6. 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Lett.76,1731(1996). 16 Penc, K. and Shiba, H. Propagating S=1/2 particles in S=1 Haldane-gap systems. Phys. Rev. B 52, R715(1995). 17 White, S.R. & Huse, D.A. Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S=1 Heisenberg chain. Phys. Rev. B 48, 38443852 (1993). 18 Kojima, K. et al. Muon spin relaxation and magnetic susceptibility measurements in the Haldane system Y2-xCaxBaNi1-yMgyO5. Phys. Rev. Lett. 74,3471(1995). Acknowledgements. Work at JHU and NIST was supported by the US National Science Foundation (NSF). Work at LSU was supported by the NSF and the State of Louisiana Board of Regents. We thank T. M. Rice, A. J. Millis and Q. Huang for useful discussions, and Rick Paul for performing neutron activation analysis.